How can I control timer555 source voltage with another timer555 frequency?
Why do you use Q2 and Q1 transistors? Use just Q1 transistor, and I think the 1KOhm resistor is not needed.
Also I think your Q1 is Upside down, NPN transistor the current flows to the direction of the arrow (From Emitter to collector)

EDIT:
Referring to the figure 2 of your new circuit:
Q1 will turn fully on when the Base voltage of Q1 is 0. If your C2 is low enough to let the Vbase of Q1 to be 0, the transistor will turn fully on, and I guess it will work then. I would recommend lowering the C2 to 10nF, or even 1nF.