Lucene, Solr, and Elasticsearch have a powerful query language and a convenient textual representation. I have seen some API providers allow users to directly use this representation within their search API. Unfortunately, when you do so you loose significant control over your development and operations. You have exposed how you process your source data for full-text searching and so can no longer make behind-the-scenes schema changes. All your queryable data must be indexed in Lucene even if doing so is questionable and/or searchable elsewhere. You are locked into using Lucene to execute the query and at scale this can be exorbitantly expensive. To fix any of these issues you end up having to break backwards compatibility. How often will your API users accept this?
Instead, design your own query language. You can base it on Lucene's syntax if you like or an S-expression as it is trivial to parse in any programming language. No matter how you express the syntax, however, you have control over the semantics and the execution. For example, perhaps one of the searchable fields is not in your full-text index, but in a relational database and its presence in the query signals a programmatic join of the indexed and relational results. You can do that. Even if your queries are easily handled by Lucene at scale you are still better off translating yours syntax, like this old-school Google query
-a +b c d
to the Lucene equivalent
not a and ( b and (c d) )
because you remain in control.
Update: I fixed the term transposition in the Lucene equivalent of the Google expression. Doh!