Real signal from I,Q, sample rate and center frequency
I gather you are trying to reconstruction a real (non-imaginary) version of your I &amp; Q signal. The following scheme should work:


Up sample your IQ from 10 MHz to 24 MHz. This involves zero padding and applying a low-pass filter to remove the extra images that appear due to the zero padding. See below for more details.
Mix the signal with $\exp(j2\pi7\times 10^6t)$. This shifts the signal to the 7 MHz.
Take the Real component of the result i.e. throw away the imaginary component.


For the upsampling process you have a couple of options. You can upsample by 12 and then decimate by 5. You would up sample by 12, apply the low-pass filter, and then down sample by 5. The filter would need a pass band from 0 - 4MHz (exact upper frequency depends on the nature of your IQ signal). The stop band would need to be at 5 MHz. You might be able to get away with a higher stopband frequency - e.g. if you have no frequency content between 4 MHz and 5 MHz, you could use a stopband starting at 6 MHz. The filter design is slightly dependent on the frequency analysis of your IQ signal and you would have to figure out your requirements for the filter specifications e.g. pass-band ripple and stopband attenuation.

For practical purposes you probably want to break the up sample by 12 into two separate steps e.g. Up sample by 4 and then up sample by 3 - each step would have its own filter applied. 

An alternative method is to up sample by a factor of 3 or 4, apply the low-pass filter and then use a spline filter to interpolate the signal to required 24MHz sampling rate.