Was the Starfleet NX class starships paying homage to Charles Lindbergh's Spirit of Saint Louis, registration: N-X-211?
Both are simply using the same naming convention, so they are related in a way, but not so directly.
The code NX was used in the USA to designate an experimental aircraft. The Spirit of Saint Louis was N-X-211 because it was the 211th X plane, the NX-01 because it was the first experimental ship with a warp 5 engine -well, actually it was the fourth, but the previous three were all considered prototypes and not experimental.
Fun fact: the use of the designation NX is completely out of place for a bunch of very different reasons:
- Enterprise is a ship, not a plane. And Star Trek usually follows more closely the navy model, so the use of a plane designation is extremely odd.
- NX is a USA thing, but there were no USA anymore when the Enterprise was launched; the Enterprise was a United Earth ship.
- But in any case, the NX nomenclature has not been in use from 1948. Yes, 50 years before the show was produced and 200 years before the time it was set in. Go figure...
NX is the registry used for experimental ships. There's no indication that it is an homage to anything specific.
In the case of Enterprise NX-01, it had the NX registry due to being the first Earth ship equipped with the experimental Warp 5 drive, along with the new transporter, and also the first to inaugurate the long-term deep space missions that would become a staple of the Federation exploration programs in later years.