Hi, GERD sufferer here. I definitely echo your symptoms of worsening acid when having a cold - even a mild one. Having heart burn is my first indicator that I might be coming down with something. I've learned to supplement my Esomeprazole (none of the others work) with ranitidine over the few days and use antacids/Gaviscon advance as needed.
Extra propping up when 'laying down' is also required. It really reinforces all the usual GERD management tips, like eat little plain meals to nourish, avoid greasy or strong flavours, avoid caffeine, avoid acidic (fruit juice) or fizzy drinks. I like to drink lots of hot water with fresh ginger and honey :-)
Hope this helps you manage things as it's horrible to lose sleep and be in pain.
I'm in line for surgery next year as my issues are mechanical rather than acid based, but the treatment is the same.
Rainbow x
I am NOT a Dr so please take this as theory.
If you are on meds I would doubt there is much association. If you've only just started them or have only been on them for 4-6 weeks there could still be inflammation and damage in the esophagus which if exposed to the flem and flu would cause the immune system to generate a lot of saliva in your esophagus as it might interpret it to be under attack.