Can I say "hit the bushes" to mean escape?

Can you use it? Yes. Should you? Maybe.

Really, you can make up or use any sort of saying or metaphor that you would like in a novel. The only basic guideline is that it should make sense. The metaphor

like a bull in a china shop

makes sense without surrounding context. You can obviously tell it goes along the lines of someone who breaks things or is insensible in a situation where you should be careful.

However, if I just made up an obscure idiom/metaphor, would you be able to decipher it? Try this:

like a stick figure in the rain

The meaning? Well, I intended to mean someone who can easily slip through precarious situations unharmed. You probably didn't get that though, you would have needed surrounding context.

So, in summary, your idiom should make sense without surrounding context in order to work.

Specifically, should you use "hit the bushes"?

Well, I for one, did not understand it without the explanation behind it, so maybe not? It's up to you. It's always best to stay simple when using idioms, so your reader does not misinterpret them. Think about it, would you need surrounding context for "hit the bushes"?

Also, be careful that no one mixes up "hit the bushes" with "beat around the bushes" as the two sound similar.

Is this phrase common?

Apparently, it's been used before. See the below chart:

-Google Ngram

But, simply googling the term yields no results. It seems like a low use idiom.

TLDR

You may use it if you would like to, but only if you are 100% confident that the reader will understand it. If you do use it, surround it with helpful context.

It also has been used before, but not super often.

Unless there were bushes, the phrase would not be understood.

See Google Ngrams for the search terms hit the bushes,hit the trail where it is used only literally (with "hit" = start/enter into) and without the meaning of "to escape" - other than where there are actually bushes.

We Were Heroes: The Journal of Scott Pendleton Collins* Walter Dean Myers · 2012 Their first wave slowed as they hit the bushes and the grenades tore into them.

Magic Apples: Reflections to Mull - Page 233 Lee Steels · 2014 Page 233 Then we used our Mother's biggest sieve to hit the bushes. If it was done correctly, only the ripe blueberries came off.

The Making of a Healer: Teachings of My Oneida Grandmother Russell FourEagles · 2014 Page 105 So everyone started making breakfast and hit the bushes for latrine duty, and at 0700 the sergeant told me to check the ambushes.

I know "beat the bushes" to describe a thorough search.

That is quite uncommon and rather old-fashioned.