Nope
it was WARCRAFT and warhammer fantasy that was a blizzard/GW thing, starcraft had a lot less influence from 40k
Plus a: s
tarcraft terrans are completely different from the Imperium, space rednecks/ muricans. Even considering the more lax approach with rouge trader they were completely different
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Zerg/tyranids: Both are inspired by HG gieger's Alien
1993: 2nd edition nids are multi-colored weird things, they were more like anthropomorphic bugs that used swords and guns.
1998: starcraft popularizes the image of the dark, creepy, insectoid spacebug
2001: 3rd edition nids drastically change artstyle to match that.
Even then, the modern depiction of the zerg are very different from 40k.
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protoss: Nothing like the Eldar design and asthetically, pre-dates the Tau, the psychic alien trope is very old
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storyline: Terrans: overthrowing governments 3 times in 7 years. Protoss have multiple collapses and reforms, zerg get humanized.