You probably would want to perform an experiment using double mutant cycles (some of the first papers describing the use of double mutant cycles to measure thermodynamic contributions of certain intermolecular interaction to protein structure and stability were done in the '80s and '90s by Alan Fersht, I believe, so you'll probably want to look up some of those papers). Here's a more recent citation from a colleague of mine that measures the strength of hydrogen bonds inside of a transmembrane protein:
Joh et al 2008. Modest stabilization by most hydrogen-bonded side-chain interactions in membrane proteins. Nature 453: 1266
PMCID:2734483Double mutant cycle analysis requires the assumption that the mutations you introduce do not introduce any compensatory interactions, so in addition to performing the unfolding experiments on the mutants, the authors of the paper also had to solve the crystal structures of the mutants in order to see that, after breaking a hydrogen bond by mutating one of the residues involve, that the other residue did not change its position significantly in order to interact with another residue in the protein.
Now, the paper I cited measures the strength of hydrogen bonds formed between amino acid side chains. You seem to be interested in hydrogen bonds involved in forming secondary structure which is more difficult because they involve the peptide backbone and not the amino acid side chains which can easily be broken by mutation. I'm not sure if there is a way to adapt the method for studying the energetic contributions of hydrogen bonds between backbone atoms (other than perhaps trying unnatural amino acid mutagenesis).