Generally, no, not without reformatting that hard drive from the old W7 system first. The W7 is licensed to the dead Motherboard and cannot be transferred by end-users (you). What you could do is backup any needed data from the old W7 drive to external media first. Next, remove whatever hard drive came in the refurbished W10 desktop and replace it with the old W7 drive. If the W10 desktop did not come with any hard drive at all, this would be one way to go. You'll then need W10 install media to install the W10 back onto the old hard drive. You may be able to make that media on the W10 desktop PC as many computers come with a Media Assistant to help you make Recovery Discs for the W10 loaded onto that PC (if it came with a hard drive in it) via removable storage media such as DVD disc or USB stick.
If that W10 desktop didn't come with a program to create Recovery Media or if the W10 desktop didn't come with any hard drive at all (that's illegal by the way!), you can phone the computer manufacturer (Dell, HP, Acer, Toshiba, etc.) and purchase the Factory Recovery Media for $29-$99 US and then use that to install W10 onto that old hard drive.
Here's the rub; if that W10 desktop was refurbished and it was built prior to 2009 as many refurbished desktops are, it might be as old as 2006 (XP-era) or 10 years old. In that case if it came with a hard drive and W10 installed on it, you can put your W7 drive into it and that would be better. As drives that are older than 5 years are very likely to fail and should be replaced anyway. If your W7 system that died of a Mobo death has a 6 year old drive, that drive is probably failing as well.
The smart thing to do would be to buy a brand new hard drive from ebay, amazon, or newegg and ditch the W7 drive or the refurbished drive that came with the W10 PC; neither of those drives are any good and have tens of thousands of hours on them. You could test them if you like, but ít would be better if you just replaced whatever drive came in the W10 desktop or if no drive replace the old W7 drive. That drive is most likely failing as well and not worth using on any other computer.
This is bad news for you since you probably don't want to spend any more money on another PC than you did already by buying the Refurbished desktop. Refurb computers often contain older drives than even the computer itself, and are often resuscitated by throwing in an old drive off the refurbishers "dead drive" box and then charging you a bunch of money for a computer with an outdated or very heavily used drive! Not a good practice, but people do it all the time. New drives are generally under $70 on ebay or Amazon, so your best buying a new one. If you paid more than $130 for that desktop online, and add the cost of having to put a $70 new drive into it to get it going, the cost of that is really $200. You can buy a brand new Dell W10 desktop at Best Buy for right around $229; less when they are on sale. So, buying that refurb desktop really isn't the bargain you think it was.
Hope that answers it.
Best of luck, <<<BIGBEARJEDI>>>