Find gas pressures of injection knowing mixture pressure without knowing volume
Of course it is a pipe and it has a infinite volume hypotactically,
To use ideal gas equations is a bit tricky with mass flow rate need to convert to molar flow rate of each gas thus need to divide by $\ce{M_w}$ molecular weight respective to each gas.
Consider $\ce{1}$ time unit where inside of pipe gets
$$\frac{Q_1}{M_{w_1}}$$
$$\frac{Q_2}{M_{w_2}}$$
moles of gas to one volume section of pipe
At one time unit according to the ideal gas equation
$$\ce{PV=nRT}$$
Gas 1 will consume $\ce{V1}$ amount of volume
$$V_1 = \frac{Q_1RT_1}{M_{w_1}p_1}$$
Gas 2 will consume $\ce{V2}$ amount of volume
$$V_2 = \frac{Q_2RT_2}{M_{w_2}p_2}$$
Total of $\ce{V}$ volume per each time unit
$$V = V_1 + V_2$$
But $\ce{V}$ is just a section of the pipe it just shifts to what ever direction the gas is flowing and keeps going for ever in the infinite pipe.
But as further the gases go away just like you said turbulence will stabilize and final temperature and pressure inside pipe will be almost equal to $\ce{T_{mix}}$ and $\ce{p_{mix}}$
When it is stabilized,
Gas 1 will consume $\ce{V'_1}$ amount of volume
$$V'_1 = \frac{Q_1RT_{mix}}{M_{w_1}p'_1}$$
Gas 2 will consume $\ce{V'_2}$ amount of volume
$$V'_2 = \frac{Q_2RT_{mix}}{M_{w_2}p'_2}$$
Total of $\ce{V'}$ volume per each time unit
$$V' = V'_1 + V'_2$$
The important factor here is that number moles of gas will shift to next stage will be the same thus molar fraction for each gas is a constant. Think as $\ce{V'}$ stage in pipe
Molar fraction can be easily calculated
$$\ce{X_1 = \frac{(\frac{Q1}{M_{w_1}})}{(\frac{Q1}{M_{w_1}})+(\frac{Q2}{M_{w_2}})}}$$
$$\ce{X_2 = \frac{(\frac{Q2}{M_{w_2}})}{(\frac{Q1}{M_{w_1}})+(\frac{Q2}{M_{w_2}})}}$$
Using the partial pressure equation
$$\ce{P_A = X_A P_{tot}}$$

what is the relation?

Refer here!
$$\ce{p'_1 = \frac{(\frac{Q1}{M_{w_1}})}{(\frac{Q1}{M_{w_1}})+(\frac{Q2}{M_{w_2}})} p_{mix}}$$
$$\ce{p'_2 = \frac{(\frac{Q2}{M_{w_2}})}{(\frac{Q1}{M_{w_1}})+(\frac{Q2}{M_{w_2}})} p_{mix}}$$
There you have it final answer but consider that its one time unit thus take only the magnitude with respective unit of mass flow rate * unit time $\ce{|Q1| |Q2|}$ symbol ` is there to show gases are stabilized.