It is my turn and I have Bristling Hydra on the battlefield and Blossoming Defense in my hand. During the Beginning of Combat my opponent uses Deadlock Trap's ability on my Bristling Hydra. Can I then play Blossoming Defense or activate the Hydra's ability to fizzle the Deadlock Trap? And does it matter which I use, or would either one work?
Yes, this works like you expect it to. You can activate the hydra's ability, or cast the Blossoming Defense in response to the Trap's ability, it doesn't matter which one you use since they both grant hexproof.
When a spell or ability attempts to resolve one of the things it does is make sure it still has legal targets. In this case your Hydra would have hexproof which makes it an illegal target for your opponents spells. Since the ability no longer has any legal targets it will be countered and not have any effect.
608.2b If the spell or ability specifies targets, it checks whether the targets are still legal. [...] The spell or ability is countered if all its targets, for every instance of the word “target,” are now illegal. If the spell or ability is not countered, it will resolve normally. [...]