Short answer: No.
Don't cold call HR to get feedback unless you know someone from the firm personally. Getting the feedback through “cold calling” is practically impossible. Consulting firms reject thousands of candidates each year - it's not physically possible for them to provide feedback especially to candidates who are rejected at the very beginning of the recruitment funnel.
There are many external factors that could explain why your application could have been rejected, such as:
- A large pool of applicants (e.g., many candidates with your background, by design not everyone can get invited for an interview)
- High quality of applicants (e.g., candidates having top schools or more experience went through)
- Diversity goals (e.g., not enough female applicants)
Last but not least, your application might have been simply not strong enough. What I usually look for in candidates for consultant roles:
- Matching “minimum” quality requirements (school, GPA, first work experience)
- Ability to communicate clearly his experience & unique qualities on the A4 page (can I see those when looking 5-10 seconds at a CV? is this person structured and can prioritize or is their experience a “laundry list”?)
- Standing out from the pool of other candidates (what have you done that nobody else has done?)
Hope these comments will be helpful and guide you into understanding what went wrong.
If you believe that there was a problem with the quality of your application then engage with an external consultant who will help you review it and reapply for your dream role!