It makes sense that the writers are vague in when the other species invented their FTL technologies. They wouldn't want to handcuff themselves to specific dates.
Dialogue in "Little Green Men" seemed to suggest the Vulcans didn't have warp drive in 1947.
Quark was probably wrong in his statements.
Are there no episodes that dealt with the events leading from Earth finding how to warp?
From the first successful test in First Contact to Earth having a small group of colonies when the NX-01 launched?
No. Nothing in Star Trek at the highest continuity level (e.g. official, filmed Trek) discusses the FC to Broken Bow era.
I don't consider novel/comic Trek canon so I will not mention it.
I wish that Star Trek came up with an official, full canon way of giving fans the complete future history of Star trek of those lost eras. We all want to have a full account of the Eugencs War; a full account of World War III; a full account of the Earth/Romulan War; and finally a full account of the early days of the Federation, of the founding members joining their fleets, militaries, soldiers, and crews into the military/exploratory Federation Starfleet.
I know there are probably books/comics covering those eras. But again, I don't consider them canon.