According to common sense, the hair is a dead cell with no nerves. The brain / body couldn't possibly know whether the hair is there or not. Therefore, the body cannot respond to trimming because it can never know it occured. That is unless you pull the root of the hair, which isn't what happens when shaving. So the answer is no, hair growth cannot be possibly affected by trimming.
However, your hair will look thicker, because like any other tissue, the tip of the hair gets weaker as it gets older. Weak hair tends to thin out, split, and generally looks unhealthy. By cutting the "old" tip of your hair you leave the healthier, younger portion, giving the impression that your hair has grown thicker.
That is an answer which based on common sense. But just in case, let us look at some references. It never hurts to get a second opinion. you will find hundreds of more sources supporting my claim by making a Google Search.
So, what it boils down to is that trimming your hair won't really make
it grow back thicker. Instead, trimming your hair will just be making
it appear to be a lot healthier.
You might remember that episode of Seinfeld where Jerry decides to
shave his chest despite warnings that it will grow back twice as
thick. Although the myth proves to be true on television, it's very
much false in the real world. If shaving caused hair to grow in much
thicker, balding men would be shaving their heads for hair loss
prevention. Children's health researcher Rachel C. Vreeman and
assistant professor of pediatrics Aaron E. Carrol put this myth to
rest: "Strong scientific evidence disproves these claims. As early as
1928, a clinical trial showed that shaving had no effect on hair
growth. More recent studies confirm that shaving does not affect the
thickness or rate of hair regrowth. In addition, shaving removes the
dead portion of hair, not the living section lying below the skin's
surface, so it is unlikely to affect the rate or type of growth.
Shaved hair lacks the finer taper seen at the ends of unshaven hair,
giving an impression of coarseness. Similarly, the new hair has not
yet been lightened by the sun or other chemical exposures, resulting
in an appearance that seems darker than existing hair."
In a word, no. It’s a myth that shaving makes hair grow back thicker,
and it does not increase overall hair growth so that you get hairier –
imagine how furry your face would be if it did. So where does this
misconception come from? Well, the truth is that when you shave your
chest the hair does tend to look thicker when it grows back. This is
because shaving slices off the tips of the hair, leaving the ends
blunt and more noticeable once re-growth occurs. You get an initial
stubbly look, too, until the hairs are a few millimetres long.