What are the prospects for quantum biology? Can non-trivial quantum effects be found that are significant in brain functioning?

I'm afraid there are still more questions than answers in quantum biology and even more so in neurobiology. There is data on the use of quantum effects by plants and a small amount of data on the same birds, hidden under the loud name "quantum compass".

While there is also Fisher's theory of the likelihood of using confusion and quantum computing in the brain, the theory of quantum consciousness is revealed to us.

The article itself is here:

https: / /m.geektimes.ru/post/282646/

Yes, they can. There is a theory that a robin (Erithacus rubecula) can hold electrons in its eye in a state of quantum entanglement and thus form a system that is sensitive to ultra-small changes in the earth's magnetic field. This feature helps the birds to navigate accurately. (physicscentral.com)

There is also speculation that the olfactory mechanism uses some quantum level effects. (wikipedia.org)

For example, here's an article that talks about how quantum principles work at the macro level:

http://geektimes.ru/post/262850/

I'll add that it would hardly be possible to form the basic elements for the development of life if nature were deterministic.

Quantum biology can potentially add a whole layer of knowledge about energy processes in living systems from the point of view of quantum mechanics, and subsequently become a theoretical basis for the practical application of such knowledge.

Quite, because quantum mechanics is applicable to a large number of functions of the brain, ranging from the work of sensations like sight and smell to the details of the work of intracellular systems.