I have this sentence:
After work, my energy completely vanishes and I don’t have any motivation to study.
But I don't know what いう means in 勉強しようというモチベ...
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I'm a Japanese but not a student of languages.
So, my explanation below may not match your teacher, textbook or mother tongue.
I think your Japanese English translation is good, however you can take:
After work, my energy completely vanishes and I don't have any motivation that I will study something.
In this translation, the Japanese word
という means the English word
any motivation that I will study something.
So, the pattern
a-SENTENCE という a-NOUN can translate like
a-NOUN that a-SENTENCE.
And there is also the pattern
a-NOUN という another-NOUN.
Using this heuristics, you will get literal translation that may be unnatural English, however you will get some hints.
In this case (と)いう is just used to give a definition of what's coming before, in this case 勉強しよう.
So, という here is explaining that 勉強しよう is the type of motivation that 湧いてこない (it's not appearing).
You can find more explanations here.