The following should do it:
var queryString: string = this._manager.metadataStore.toQueryString(query);
manager being your EntityManager and query being the EntityQuery.
Result
SomeEntity?$filter=FirstFilter%2Fany(x1%3A%20x1%2FLicenseTypeId%20ne%2016730)&$expand=FirstFilter%2CFirstFilter%2FPartners%2CAddresses&$inlinecount=allpages
and here is an alternative that I don't quite like as much:
var anotherway = query._toUri(manager.metadataStore);