So sorry to hear about your stressful visit. Sounds challenging. I am so glad you are on top of it.
Are you familiar with dog body language and calming signals? If not, I would strongly suggest that you do a quick internet review of them so you can better monitor the situation with the two dogs and de-escalate quickly so no issue arises.
Dog body language to another dog can be very quick and/or subtle. It can happen so quickly and if you aren't familiar with the signs you can easily miss the warnings.
Perhaps your Mom's dog sent your dog some body language and your dog reacted upon it?
Many times we dog owners never catch the aggressive signs until it goes to a stare or bark or growl or lunge.
But if you watch carefully you may see the other body language such as your dog gives a lip lick, head turns away from other dog, dog moves entire body away, yawning, whale eye, body freeze, ears back, etc.
And perhaps I can see that maybe your mom's dog is a bit defensive or worried about other dogs coming super close since he was just in a dog fight. If I was in a fight recently, I would be worried or a bit uneasy, wouldn't you? So maybe Mom's dog is sending some subtle signals for your dog to give him space. When your dog sniffed him, did he look happy and relaxed about it?
You are right, the staring is indeed worriesome. I would immediately just take your dog away gently with a leash into another room if possible and distract with a bone, training moment with treats, etc.
Or ask both dogs to do a trick/fun command if they are good like this together and then give them some praise and treats for complying. So basically redirect their attention back to you. Turn the negative into a fun, positive, relaxed situation if you can.
Don't know if this will help you, but I know my three dogs give each other body language/calming signals all the time. And they do it to me too! Like today when I wanted to hug Puma, she said no thanks by turning her head away from me. When I asked her again for a hug she actually moved her body a bit away from me as if to say, "Ummm, no Mom, I'm not in the mood for a hug right now." I was clearly annoying her at that moment, so I stopped and just asked for a kiss--- and she said ok to that. I rewarded her with some doggie beef jerky and all was good