You may use a clock at 4 times the desired frecuency, feeding a 2 bit binary counter (use a 4 bit counter and connect Reset to Q2 to make it work like a 2 bit one).
Q1 will be one of the outputs.
To make the other output you must combine Q0 and Q1 with an XOR gate (exclusive or)
Of course, you may use a microcontroler instead... But it will need a second power supply (5V instead of 15V), and level translators on its outputs. With only 3 chips (oscilator, counter and Xor gate) it will be better to use the "discrete" aproach.
If your counter needs an "active low" reset pulse, use one of the unused Xor gates to invert it (connect one input to Q2 and the other to +V, the output will be "NOT Q2")
Don't forget to connect the gate's unused inputs to +V or ground.
sounds like you are wanting to make a quadrature clock.
you can use a sine wave oscillator, and differentiator, and comparator to get 90d shifted clocks.
likewise you can take a double frequency clock, 3 inverters, and 2 registers
use a register and inverter as a clock divider (D = not Q). invert the clock and feed a second clock divider.
if you have a clock input that you need to convert, you can use a PLL with a clock divider in the feedback path to get a 2x frequency clock.
You need a two stage Johnson counter, also called a folded ring counter.
The only caveat is that the standard CD4013 is rated at 15V maximum. You could run it at lower voltage, then add some discrete circuits for the 15V outputs.
What are you actually trying to do?
That's true but the absolute maximum rating is 18V so running it from 15V continiously won't do it any harm, even if they are slightly higher voltage spikes on the supply line. https://www.cedmagic.com/tech-info/data/cd4013.pdf