Is it possible for these two bears to mate and produce future generations?

All bears are in a single family, Ursidae. Species names are binomens, like Ailuropoda melanoleuca, so it is not correct to say "genus species Ailuropoda melanoleuca." The correct species name is Ailuropoda melanoleuca. The genus name is Ailuropoda. The name melanoleuca is not a species name but a trivial name or epithet.

The panda and the grizzly never mate because they inhabit different geographic areas and therefore never encounter each other. It is very rare for animals in two different genera to mate with each other even in captivity.

It is extremely unlikely to occur in nature. The biological definition of species is whether two individuals can reproduce and produce viable offspring that are themselves able to reproduce. The panda and grizzly are not just different species; they are different genera.

It might be possible if gametes were extracted from both and manipulated in the lab, and then one of the females of one of the species were used as a surrogate mother to implant any fertllized eggs that resulted.

Well, in theory it may be possible. If it would be possible, it definitely would never happen in the wild though.

Many animals can produce hybrid offspring. In captivity people frequently made lions and tigers breed with each other. As well as horses and donkeys. But this hybrid offspring is always infertile, meaning it can't produce future generations.

Within the bear family: I have knowledge of polar bears (Ursus maritimus) and Grizzly brown bears (Ursus arctos horribilis) being able to produce offspring. Wether or not a panda would be able to breed with a Grizzly is not known. I'm pretty sure a panda would never mate with a grizzly bear. First of all, pandas don't successfully breed often in captivity, even with their own kind. Imagine the hassle to make them breed with a grizzly. Second, their lifestyles are too different! Pandas eat bamboo and only bamboo and are obviously adapted to eating bamboo, while a grizzly is an omnivore with adaptation of a predator. What would the offspring eat? It just wouldn't work.

While in theory this is feasible, it just can't happen for real.

The definition of species usually says that two animals are the same species if they can have fertile offspring. Animals of different species but the same genus can have offspring that are not fertile, like the horse and donkey. The bears you name are the same family but different genera, indicating a divergence of similarity.