It didn't look or sound like one of the Hope and Crosby road movies to me, but then I haven't seen them. None of my usual online sources had any trivia on this one.
Here's the dialogue from that scene, courtesy of Chrissie's Transcript Site:
PHLOX: Something tells me we haven't seen the last of the detective in the bow tie.
TUCKER: No, he died in a house fire.
PHLOX: Ah, did he? The autopsy was inconclusive. I wouldn't be surprised if the body belonged to the delivery man with the strange limp. You never did see him leave the house.
TUCKER: What about the gardener? He was there too.
PHLOX: Too tall. Even the primitive forensics of the mid-twentieth century would have determined that.
T'POL: Perhaps we should watch, and find out.
TUCKER: Part of the fun of a mystery is trying to solve it before it ends, using logic. You of all people should appreciate that.
T'POL: Then use logic more quietly.