Suppose your (first) Quarter on Quarter growth rate was 3% and that spanned 3 months and you want to know how much each month grew. That is you want to know the growth rate for Jan, Feb and Mar, call them $\alpha, \beta, \gamma$.
The only information you have is that:
$(1+\alpha)(1+\beta)(1+\gamma) = 1 + 3\%$
This is one equation for 3 unknowns and therefore has 2 degrees of freedom. You have an unlimited number of potential solutions.
One possible solution..
If you choose to make the assumption that the growth rate in each period is the same then then you have 1 equation for 1 unknown, and this implies that:
$(1+\alpha)^3 = 1 + 3\% \qquad \implies \qquad \alpha = 0.99\%$
A second possible soultion..
If you knew that December's growth rate was, say, 0.4% and you assumed there was linear increase in growth across all 3 months this would form a different set of equations:
$(1+\alpha)(1+\alpha+x)(1+\alpha+2x) = 1 + 3\%$
$\alpha-0.4\%=x$ (this is the change from Dec to Jan)
This implies that: $(1+0.4\%+x)(1+0.4\%+2x)(1+0.4\%+3x) = 1+3\%$
and $x = 0.295\%$
so under this assumption the growth rates in Jan, Feb and Mar are 0.695%, 0.99% and 1.285%.
Basically you can't create information from nothin so you have to form your own assumptions.