length function of a coxeter group with respect to two different simple systems are equal or not? Is there any relation between length function of a coxeter group with respect to two different simple systems as two simple systems are weyl conjugates of one another?
 A: The answer to the question in the title is "no". Let $W=S_3$. Then the transpositions $\{ (12), (23) \}$ form a simple system and $\{ (12), (13) \}$ form another. The length of $(13)$ is $3$ in the first system and $1$ in the second. 
As to the question "is there any relation?": Well, they are equal mod $2$. Also, as Lee Mosher says, they are within a constant factor of each other; that fact is only interesting for infinite Coxeter groups.
I suspect that there isn't any more to say, but it is always hard to be sure that there isn't some other relation between two things.
A: To supplement what David Speyer and Paul Garrett have written, I'd emphasize that the question itself is faulty: the terminology involved in "two simple systems are weyl conjugates of one another" doesn't make sense here.   Whenever one speaks of a Coxeter group one is implicitly referring to a pair $(W,S)$ consisting of a group $W$ and a specified set $S$ of generators, along with certain explicit relations among those generators.   Only when you fix this data do you get a definite length function (relative to $S$) for $W$.
The tag "lie-algebras" already indicates some fuzziness here: there is no Lie algebra and no Weyl group.  Appropriate tags include "gr.group-theory" and "coxeter-groups"  As David's example indicates, even for simple Lie algebras and their Weyl groups, the Weyl group as a Coxeter group may admit multiple unrelated presentations.  And even if one requires $S$ to be a set of simple reflections in the sense of Lie theory, any root can be a simple root and have length 1 relative to some choice of $S$.   So I'm not sure what the question is really about(?) 
