Is the following sentence grammatically correct?
I began to notice that things were suspicious.
If I google "that things were suspicious" it only comes up with 5 things.
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Your sentence is grammatically correct -- there is nothing wrong with the word order, but it is not semantically correct. Like Lewis Carroll in Alice in Wonderland, writing "the slithy toves outgrabe" it has good structure.
But what does it mean "things were suspicious"? How can "things" be capable of suspecting?
You could say, "things looked/felt suspicious" and it would be a little more clear.
Are you trying to say, "I began to feel suspicion"? In that case, use the active mode and and say, "I began to suspect..."