Using a standard trick to include a single quote:

perl -lane '$F[1]=~s/(chr\w[^_]?)_.*\b/$1/;print $`. "\t". $F[1] ."\t"
+ .$'"'"
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The single quote directly following the $ matches the first single quote. In Perl, this will end a string, but not so in the shell - here single quotes act like \Q/\E in Perl. Directly following the $ and the single quote is "'", which for the shell is just a single quote. Which makes that the Perl program ends in $', as intended.