I don't know specifically regarding lovebird hybrids, but speaking in general, the fertility of hybrids varies greatly, depending on the species involved. For some combinations, all hybrid offspring are sterile or nearly sterile, so the possibility of a second generation of hybrid offspring is unlikely or impossible. For other hybrids, their fertility is not affected or minimally affected, so it is quite possible for the hybrid to produce viable hybrid offspring.
Mules are an example of a sterile hybrid, produced by breeding a horse with a donkey. This is likely due to the fact that horses have 32 pairs of chromosomes and donkeys have 31 pairs of chromosomes. Mules end up with 63 chromosomes (31 & 1/2 pairs) and that extra mismatched chromosome interferes with viable reproduction. This is not the only issue. There are also other differences between horse and donkey genetics which produces reproductive problems. Even though they are relatively similar, they are still different species and these differences can cause serious issues that lead to fetal death. This is actually one of the reasons why hybrids themselves are usually uncommon. Many times, even if two different species breed, the hybrid offspring dies before being born. The hybrids that actually make it are the exceptions. Interestingly, even in mules, there are a few rare cases of viable offspring being produced. It is quite rare, but it does occasionally happen. In all cases, the mother was a mule and the father was either a horse or donkey. I couldn't find any examples of a mule stallion producing offspring or of two mules conceiving.
From what I've read regarding parrots, first generation macaw hybrids are usually able to conceive and have viable offspring. The likelihood of success is greater when one of the parents is not a hybrid. It is lower if both parents are hybrid. And fertility tends to go down with more crosses. So a second or third generation macaw hybrid is more likely to be sterile than a first generation hybrid. Most of the stuff I found was anecdotal, since hybrids have not been closely studied. And they are often the product of accidental or poorly-planned breeding, so record keeping might be a bit dodgy. I would guess that crossing closely related macaw species would be more likely to produce fertile hybrids and crossing more distantly related macaw species would produce more sterile or low-fertility offspring. But that is just a guess. I don't think anyone has tried to test that theory.
It also suggests that too much intentional hybridization could negatively impact macaw populations in aviculture due to increased genetic incompatibility and lower overall fertility. Probably not an issue in the immediate future, but something to think about for future generations of parrot owners.