Is metallicity low at the central region or nucleus of the Milky Way? Does metallicity decrease or increase as we move from the center out to the edges of our galaxy?
The stars in the Galactic bulge are predominantly metal-rich (by that I mean have a metallicity similar to the Sun or even a little higher).
Even though these stars are predominantly old, the bulge is thought to have formed extremely quickly and the interstellar medium from which the stars were formed would have been enriched with metals very quickly.
Here is a plot from Zoccali et al. (2009) (which I recommend reading). It shows the metallicity distribution of many stars in the Galactic bulge, measured using high-resolution spectroscopy. It shows that the highest metallicity stars are towards the middle (it is very hard to get samples right in the middle because of extinction) and the averge metallicity falls as you mover further from the centre (the samples with more negative Galactic latitude $b$.). The percentages in the plots are the estimated contamination from the disk population in each sample.
The dependence of metallicity on height and radial coordinate in the Galactic disk is still keenly debated. There is general consensus that the metallicity falls with radial distance from the Galactic centre and with height above the Galactic plane. The gradient is of order a few hundredths of a dex per kpc between a few kpc and 10-12 kpc from the Galactic centre. It may flatten beyond this. You could have a look at the discussion in Cheng et al. (2012), but there are many other attempts at parameterising the gradient and it is a technically difficult thing to do.
1$\begingroup$ That Zoccali paper sure does start off humbly. "Our brave team of astronomers was not daunted by these difficulties.... In this tour de force we measured hundreds of bulge stars..." $\endgroup$ Jun 22, 2015 at 4:14