If indeed you hooked up to city water and not tank flush, I'm going to bet that something in the RV isn't plumbed properly.
If everything is plumbed correctly, the only way water would get into your grey tank is via the kitchen/bathroom sink/shower faucets to their respective drains, which I think you would see/hear. Or the toilet, which would have to fill the black tank before getting into the grey (which on some units isn't even possible), which I think you'd also see/hear.
Please be sure to come back and tell us what happened.
1988 Coleman Sequoia - popup (1987-2009) - outlasted 3 Dodge Grand Caravans!
2012 Roo19 - hybrid (2012-2015)
2016 Mini Lite 2503S - tt (2015 - ???)
2011 Traverse LT, 3.6L, FWD
2009 Silverado 1500 Ext Cab, 5.3L, 4x4, 3.73
2016 Silverado 2500HD Dbl Cab, 6.0L 4x4, 4.10