What resolution is needed to see a quasar through the galactic center? The center of the galaxy is densely packed with stars and obscured by a whole lot of dust between us and it. For those reasons, groups that study the motion of stars around the super-massive black-hole in the center of the Milky way need to use big telescopes at near infrared wavelengths. Questions of dust obscuration aside, what resolution would a telescope need to have to pick out a background quasar looking through the center of the Milky Way?

SDSS QSOs start at around 8th magnitude (Vega) in 2MASS photometry ($H$ and $K_s$ bands), and rise like a power law as flux goes down Here's a quick and dirty plot of SDSS QSO survey made from SDSS data release 10 made using topcat. I would convert the graph to counts per solid angle, but I'm too lazy to look up the survey area of the SDSS spectroscopic survey right now. 


There's a quick and dirty example of the effect I'm describing in Figure 2 of the Ghez et al. (2008) paper Rob Jeffries linked to below. The difference between the red points and blue points is both exposure time (more photons) and higher Strehl ratios (a measure of how close to diffraction limited the image is and, hence, a proxy for resolution given the same optical system).

Generically, you could say that the question is: how do we determine the confusion limit in images with point sources? This is just specifically applied to the field of view on the sky with the highest density of resolvable point sources available. Concretely, say we wanted to resolve $22^{\mathrm{nd}}$ magnitude sources in views of the galactic center, what resolution is required to do so?

Getting to the full answer to the more concrete question "how deep would we have to look to see a background quasar" would require additional information: size of field of view, and dust extinction along the sight line. So, for simplicity, I'm assuming that if you're resolving $22^{\mathrm{nd}}$ magnitude sources in $K$ band, you're probably seeing a quasar. Thus we only need the projected density on the sky of sources brighter than that cutoff, roughly, and some description of how resolved sources need to be from the background brightness to detect them to answer the question.
Whilst awaiting clarification, I'm going to assume that your question is asking about calibrating the reference frame within which the motions of stars at the Galactic centre are measured. 

The process is described by Ghez et al. (2008); does not involve observing quasars at infrared wavelengths; and the study does not suffer as a result.

The Galactic centre adaptive optics images are 5-10 arcseconds across, but contain thousands of stars. Only a fraction of these are close to the Galactic centre. The ensemble can be used, iteratively, to define a relative reference frame by minimising the displacements of thousands of stars, excluding those very close to Sgr A* and those with large motions.

This is really all that is required to do the orbital analyses of the black hole.

To put the coordinates in the International Celestial Reference System, a wider field is observed that contains a few infrared-bright giant stars, that are also bright maser sources at radio wavelengths. The positions of these are used to bootstrap the coordinates onto the ICRS, and this is important if, for example, we are interested in the relative motion of the Sun around the Galaxy.

The maser sources have ICRS positions by virtue of their radio source positions versus the radio coordinates of distant (but still bright and point-like) background quasars.The issue of background sources, extinction and confusion does not arise at radio wavelengths.