Budgies are opportunistic breeders, so if conditions are right, they can stay in breeding condition for a long time - practically year round. On the flip side, if conditions don't favor breeding, they might not go into season for years. It can be quite variable.
There are a few simple things and some more involved steps that you can take to discourage breeding, if the biting is hormone-related.
A great first step would be to rearrange the cage, along with adding more fun toys, and REMOVING any nests, boxes, or enclosed nest-like toys (coconut shells, snuggle dens, etc). Chew toys and shredding toys are usually well-received at this time because female budgies have a natural urge to carve out a cavity for nesting. Just avoid providing an actual cavity or you could end up with some eggs. This might happen anyways, but since we are trying to reduce hormonal behavior, it is counter-productive to provide an ideal nesting location.
Additional fun distractions might help with the biting even if it is not hormonal, but there are many reasons why a bird might bite. The first step in solving a biting problem is usually narrowing down the triggers so you can work out WHY or at least WHEN the bird will bite. With that information, you can work on an effective plan for avoiding or redirecting the behavior.