I usually find if I get that kind of output it's because I haven't given enough information in the form of limits, i.e. also include an upper value for {l, m, n, o, p, x}. However, if you examine the behaviour of just the first Mod section you will find it is cyclic about these numbers
{122/9, 212/9, 302/9, 392/9, 482/9, 572/9, 662/9, 752/9, 32/9, 122/9}
and you are asking for Integer solutions therefore it wont find any. You could try removing the ==x so it just compares the Mod sections, you would still need to include an upper limit for the variables and in this case it would run for hours even with the limit set at 90. Simplifying your equations somewhat and including an upper limit just for x returns a result quite quickly.
Solve[Mod[(14 + (90 l + 18))/9, 90] ==
Mod[(1 + (90 m + 45))/9, 90] == Mod[(46 + (90 n + 90))/9, 90]
Mod[(80 + (90 o + 54))/9, 90] == Mod[(83 + (90 p + 27))/9, 90] ==
x && x < 90, {l, m, n, o, p, x}, Integers]
{}
Returning to the limits it's also wise to include lower limits too as in 0 < x < 90 if you are interested in just positive solutions.
