Are these components connected correctly?
in short: Definitely Not.

1 : you do really need to connect the trigger pins of the ultrassonic sensors, or else you will not be able to "trigger" them.
Read the ultrassound library examples, they may help you.

2 : I am pretty sure an analog (or digital for that matter) pin in arduino nano cannot sink the current needed to make the motor spin. You are going to fry that pin/port or even brick the entire arduino.
Read on power mosfets for the solution of that.

Moreover, i don't think the 5v out of the arduino feeding the motor is a good idea as well, as the current given by its internal regulator may not be enough for the motor as well.