Yes, you can apply deep learning to peak detection.
A 1D CNN would be appropriate for this task.
Here is an example for such application:
Risum, Anne Bech, and Rasmus Bro. "Using deep learning to evaluate peaks in chromatographic data." Talanta 204 (2019): 255-260.
You would need to have annotated data.
If you decide to stick with the classical ...
I think the most straightforward basic approach to detecting noise/variance levels is:
highpass -> rectify (or square) -> lowpass
There are many types of 'highpass' or 'lowpass' like operations/filters that you can choose from and tune based on your application. Below I use diff (e.g. derivative) effectively as a highpass filter, and a couple rolling ...