In fact, there are a number of kindergarten screening instruments that include measures of social-emotional-behavioral development, which may be considered indicators of maturity. The most prominent (and strongest, in my opinion) among which would be the DIAL-4. There is a quicky screening version of it, which takes about 5 minutes all told, and generates a single global score, interpreted as making or missing the cut score, and there is a more comprehensive version, where each of the mandated areas of potential developmental delay is assessed (about five minutes apiece). The social domain can be administered as a stand-alone measure. So if the school wants somewhat objective, quantitative data on "maturity", this would probably be about as good as it gets.
OTOH, I think what indigo and CD mentioned about inquiry would probably be more to the point, as "maturity" is so often a code word for "we're reluctant to do this, either because there is institutional bias against early entry, or because we're afraid we won't know what to do with her". When we entered our first early into 1st grade (skipped K), it took ongoing discussion and assessment from April until the end of August to attain consensus. And this was with a school with which I personally had a long history of warm and professional relationship. In our case, staff were genuinely concerned that this might be to the detriment of our child; we had to work through their fears and concerns, as much as demonstrate academic appropriateness.
