Modify the content of variable using sed (or something similar)

You can do pattern substitution directly in bash:

P_REP=${P/./_}
Q_REP=${Q/./_}

From the bash(1) man page:

Paramter Expansion

${parameter/pattern/string}

Pattern substitution. The pattern is expanded to produce a pattern just as in pathname expansion. Parameter is expanded and the longest match of pattern against its value is replaced with string. If pattern begins with /, all matches of pattern are replaced with string. Normally only the first match is replaced. If pattern begins with #, it must match at the beginning of the expanded value of parameter. If pattern begins with %, it must match at the end of the expanded value of parameter. If string is null, matches of pattern are deleted and the / following pattern may be omitted. If parameter is @ or *, the substitution operation is applied to each positional parameter in turn, and the expansion is the resultant list. If parameter is an array variable subscripted with @ or *, the substitution operation is applied to each member of the array in turn, and the expansion is the resultant list.

John Kugelman's answer is fine for your example, but if you need to process the content of a variable with the actual sed program (or some other arbitrary command), you can do it like this:

P_REP=$(sed 's/\./_/' <<< "$P")