I've been using a MultiTech MT5656ZDX - V external Fax/Voice modem with a Serial cable connected to a USB adapter cable plugged into a USB Port for nine years.  The program I use to answer the phone and fax in/out of is VentaFax.  My OS is Win 7 32-bit.

With no warning a few weeks ago, the modem can see VentaFax when it's activated, but VentaFax gives me an error message. It can't recognize the modem.  I've tried querying the modem with the Windows Fax & Scan program. Same results. To be sure, I downloaded a new 15-day trial Fax/Voice program to test the modem in case the problem is the program.

During the installation, it queried the modem and said it was busy with another program which it was not. This cancelled the installation.

I hate to purchase a new modem if I don't need to, but it is 9 years old. I've considered completely uninstalling the Modem drivers and reinstalling them again, but if the modem is bad, I'll be wasting my time.

Any suggestions anyone? I'm a newbie here and would sure appreciate some help.This is from dictation; please overlook the obvious errors.
You'll find many people here willing to help you and that understand the kind of problem you're having. You say your new be, so I don't want to throw a lot of tech stuff that you and besides would probably would not help very much anyway.
I've been working with electronic technology since I can remember, but I'm not really a genius, I just have lots of experience and know a lot about a little bit. Things like facts machines and other gadgets sometimes will fail to work with Windows for reasons that it's hard to understand.
Let me make a comparison or what they call a parable. Imagining you're going fishing and you have the choice of fishing from a small pond or from a fast-moving Creek. It's going to be quite different. In the pond you know there's a limit about a fish available and once you've caught a number of them there might not be anymore left or if they are there hiding from you. But a fast-moving Creek is different. It is a stream and there could be always more fish coming down stream at you and if you miss a fish three gets away from the you probably keep on going downstream and you never see him again. Okay. This is help you understand the idea of a stream device. In your computer many external devices are considered to be stream devices because the information can change a lot and you can go back and recover it or get it back if you lose it. That's one of the definitions of a stream device. Well, sort of. It is just a METAPHOR but it conveys the idea of something that cannot easily be shared.
What this means is that some programs are mutually exclusive when it comes to sharing stream devices a stream device like a microphone or a fax or printer can only be used by one program at a time because it is a stream device. This means the program might exclude or shut out any other program that wants to use the device at the same time. This can be a real nuisance if you have the kind of program that stays in the BACKGROUND just laying to see if the stream devices become active.
Okay, I think I told enough you get the idea. You'll probably have to completely uninstall all the software drivers and programs related to the fax modem and just go get a new fax modem.
Why new? Because is no easy way to tell if your nine-year-old equipment is still in a GOOD. Maybe it is. One way to find out is to go get a completely new computer and see if the old equipment will work on a completely new computer. Of course, that is rather extreme. You would not do that as an individual but only if you are some kind of a corporation and you had to know.
Otherwise, as an individual, the most cost-effective thing is just to get a new modem and do a complete new installation of the software.
I'm trying to give you a quick answer and this may not be the answer that WORKS for you. I'm just giving you my quick opinion. I fooled around the stuff for years and years I've had this happen to me that I will give up on something and go buy something brand-new and then the old device that failed start to work again. That's just the way it is.
I HOPE this is of some help to you.   
There is several variables in your configuration which need to be isolated. There is the USB to serial adapter which must include electronics to do the conversion which may have failed. You can check this if you list what lights are on on the modem. The lights should be quite bright, if they are dim it could be the 9v plug pack is failing.
This is from the modem manual https://www.multitech.com/documents/publications/manuals/s000248m.pdf

Power-On Test
Test the modem by turning it on (an on-off switch is located on the side panel). When you apply power, the
modem performs a diagnostic self-test. The PWR indicator lights; and if a terminal program is running, the TR
indicator also lights. If this does not happen, check that the power switch is on, the power supply is solidly
connected, and the AC outlet is live. If these measures do not work see the Troubleshooting chapter.

Next check windows device manager for any exclamation marks. What driver is installed for the usb to serial adapter if it is a generic windows driver then try rolling back the driver.

Check your antivirus isn't blocking USB data flow. By shutting down the antivirus temporally.

If all of the above checks out then it could be windows setting up the usb modem support incorrectly. You can delete the USB support and re install with programs like USB oblivion https://www.cherubicsoft.com/en/projects/usboblivion

There are still USB dial up modems with fax support available if the modem is faulty. These are not a patch of the quality of the Multitech you have so would be worth doing a bit of diagnostic first before writing it off.
I suspect the 9 v Multitech power supply as the most likely cause of the problem these were early versions of switch mode power supply plug packs and failed if mounted horizontally. Like in a power distribution panel not a socket on the wall.