Probing the electronic state in the strong topological metals Zr$_{2}$Te$_{2}$P, Hf$_{2}$Te$_{2}$P, and Zr$_{2}$Te$_{2}$As and their structural analogue Ti$_{2}$Te$_{2}$P*

Recent work shows that Zr$_{2}$Te$_{2}$P is a strong topological metal with time reversal symmetry protected surface Dirac cones at the and points (in the hexagonal basis), bulk bands with a possible Dirac-like character at the point, and conventional (quadratically dispersing) bulk electronic bands centered around the point. We report results for the broader family of materials $X_{2}$Te$_{2}M$ ($X=$ Ti, Zr, Hf, and $M =$ P, As), where we find remarkable agreement between the calculated electronic states and those measured through quantum oscillations in the magnetic torque. Analysis of the quantum oscillations additionally reveals small ($m$* \textless 0.1 $m_{e})$ charge carrier masses and non-trivial Berry phases for the Fermi surfaces at the point, and more conventional behavior for the bands that are centered around the point. These results will be discussed with attention to the relationship between topologically protected surface states and bulk metallic conduction. We also discuss prospects for designing new materials in this structure.