No- they don't. Only certain elective deferrals qualify for the Retirement Saving's Credit. 414(h) contributions are not deemed as elective employee contributions for this purpose- they are deemed as employer, not employee, contributions (aka "pick-up").
From the Form 8880 instructions:
Contributions designated under section 414(h)(2) are treated as employer contributions and as such they are not voluntary contributions made by the employee. They do not qualify for the credit and should not be included on line 2.
https://www.irs.gov/pub/irs-pdf/f8880.pdf
https://www.irs.gov/government-entities/federal-state-local-governments/employer-pick-up-contributio...
