While the state the highway patrol is located in is never officially disclosed, the map Mathews refers to after receiving the bomb call shows US routes 34 and 36 along with towns named Akron and Brush. This is an actual location in northeastern Colorado. Colorado has a state patrol, not a highway patrol.
As an indication of his subsequent, and widely acknowledged, attention to detail; writer Gene Roddenberry, as Robert Wesley, references the 1947 Texas City port explosion, which destroyed much of that Galveston county town.
Early episodes reportedly used real CA Highway Patrol Cars and this appears to be the case here. It has a rare interior shot of Matthews car (1955 Mercury) where you can see the two way radio Dan is always using. The microphone is mounted high on the edge of the dash and the control head for the radio is mounted underneath the dash just below where the regular AM radio would mount. In this case the car is a "radio delete" where it never had a factory AM radio installed and instead there is a block out plate in the dash. A photo of this is in the photo section for this episode.
Suspect Jay Detterick is driving a 1954 Plymouth Plaza Suburban 2 door Station Wagon. It appears to have a front license plate of a California Commercial vehicle, which utilized the 1951 base plate. The letters "COM" are stacked on left side of plate, then followed by "515795." The plate color would be black and the character color was chrome yellow. The year of registration was a steel tab with a red background. The white number "55" or "56" would have been at the lower right corner of the rear plate. "19" would be under the stacked "COM" on the left, then "California" under the numbers and then the year "51" on the right which the steel tab covered.