star trek - Did Starfleet ever have an Enterprise-class of starship?

In canon, there has never been an Enterprise-class starship in Starfleet.

But a lesser-known fact is that the decision for the Enterprises NCC-1701 and NCC-1701-A to be "Constitution class" was not made until 1987. Before that time, no class designation was given in dialogue in TOS or the films before that year. Prior to 1987, both "Starship class" and "Enterprise class" were used informally amongst Star Trek creative personnel to describe the original Enterprise, and "Starship class" appears on the bridge dedication plaque on the NCC-1701. ("Enterprise class" was specifically floated by Andrew Probert for the refit Enterprise in The Motion Picture, but this idea was dropped.)

Also, in the original script for The Wrath of Khan, the Reliant was described as "Enterprise class", and indeed the ship was intended to be a copy of the Enterprise. However, it was decided at the design stage to fundamentally change the shape of the Reliant in order to better distinguish the two ships in battle. Later, the Reliant was retconned as "Miranda class".

For a history of the complicated evolution of "Constitution class" from "Enterprise class", with reliable citations and direct quotes by Enterprise designer Andrew Probert, see here.