The inequality as stated is false, but it is true that $$\langle v,x\rangle^2+\langle v,u\rangle^2\leq 1+|\langle x,u\rangle|.$$ Moreover, the right-hand side is optimal in the sense that it is the maximum of the left-hand side over the unit vectors $v$. More generally, if $u_1,\dots,u_R$ are any vectors in a Hilbert space, then $$\max_{\langle v,v\rangle=1}\sum_r|\langle v,u_r\rangle|^2$$ equals the largest eigenvalue of the Gram matrix $(\langle u_s,u_t\rangle)_{1\leq s,t\leq R}$. For details, see these notes on the Bombieri-Halász-Montgomery inequality.

The inequality as stated is false, but it is true that $$\langle v,x\rangle^2+\langle v,u\rangle^2\leq 1+|\langle x,u\rangle|.$$ Moreover, the right-hand side is optimal in the sense that it is the maximum of the left-hand side over the unit vectors $v$. More generally, if $u_1,\dots,u_R$ are any vectors in a Hilbert space, then $$\max_{\langle v,v\rangle=1}\sum_{r=1}^R|\langle v,u_r\rangle|^2$$ equals the largest eigenvalue of the Gram matrix $(\langle u_s,u_t\rangle)_{1\leq s,t\leq R}$. For details, see these notes on the Bombieri-Halász-Montgomery inequality.