Irrationally, Can This Happen?

Number Theory Level 4

True or False?

The infinite sequence $\big\lfloor 1\times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 2 \times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 3 \times \sqrt{ 2017 } \big\rfloor,\ \big\lfloor 4 \times \sqrt{ 2017 } \big\rfloor,\ \ldots$ contains infinitely many perfect squares.

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.