We consider the triangular q-summability of 2-dimensional FOURIER TRANSFORMS. Under some conditions on q, we show that the triangular θ-means of a function f belonging to the Wiener amalgam space W(L1,l¥)(R2)W(L1, l¥)(R2) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points for the so-called modified Lebesgue points of f∈W(Lp,l¥)(R2)f∈W(Lp,l¥)(R2) whenever 1<p<¥1<p<¥. Some special cases of the q-summation are considered, such as the Weierstrass, Abel, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.