<p>In this paper, two dimensional generalized Boussinesq equation <Equation ID="Equ28"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1322_Article_Equ28.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="342" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} u_{tt} -\Delta u +\Delta ^2u+ \Delta ( u^5 ) =0, ~~~~~~x\in \mathbb {T}^2,~t\in \mathbb {R} \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mi>u</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mn>5</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mspace width="3.33333pt" /> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>under periodic boundary conditions is considered. It is proved that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem, partial Birkhoff normal form and scaling skills.</p>