I can definitely see how a genuine libertarian could be a Trek fan.
The politics of Star Trek is all about individual dignity and fulfillment in a post-scarcity society. A lot of people try to call it socialist (as Pelia mockingly did in the most recent SNW episode) but the circumstances mean it's not any form of socialism anyone's encountered in real life on Earth, such as in the 20th century. After unfathomable levels of technological advancement eradicates the problem of scarcity, there's neither the need for a big state nor a market to allocate scarce resources - what we know as socialism and capitalism wouldn't be meaningful concepts. What we see instead is people doing what they do (joining Starfleet, undertaking research, conducting journalism, opening restaurants) out of a sense of personal fulfillment, and with neither a state nor a society nor a need to pay the bills particularly forcing them to do anything. They're free to live their lives as they see fit - infinite diversity in infinite combinations. I can see how a libertarian could look at that and call it their personal utopia.
I struggle much more with how a conservative could embrace Star Trek. So much of conservative politics is about the primacy of the norms of the collective over the rights and dignity of the individual - whether that's in moderate forms (e.g. wanting to manage the pace of social progress so as not to offend the sensibilities of the majority, wanting immigrants to integrate into host societies) or more aggressive forms (outright hostility to immigrants, denying the rights of women and minorities, denying the existence of LGBTQ people).
I guess what I'm saying is that once you remove economics from the problem of politics (as Star Trek has hand waved away via technology) then what's left of libertarianism looks a lot like Star Trek, whereas what's left of conservativism looks very different.