There is a taboo about using double negatives in English, but that only applies if you need one negative and you use two. For example, if you wanted to say that you intended to remain where you are, you could say:
I'm not going anywhere - correct
I'm not going nowhere - incorrect - double negative
There are, however, situations where two negatives are required to convey the correct meaning. Say somebody intends to skip an important social occasion, one might correctly say
You can't not go!
Using 'no doubt' before a negative is fine, because it simply emphasises the truth of whatever follows.
No doubt [there is] not a single bird nest in this tree.
Looking at your sentence, it could perhaps do with a verb, but otherwise it's fine.
Note that no doubt is generally used to reinforce an opinion or prediction, rather than something that is easily verifiable. Here is a more realistic usage:
No doubt there won't be a single bird left in this tree by the time you get your camera out.