"The deep" can mean "the sea."
This use is literary and old-fashioned.
Poetry is never easy to understand, and this is no exception. I'm not sure exactly what is meant in a literal sense. He is saying that somebody's breast, i.e. their chest, is not heaving, i.e. someone is calm. I can't tell if he is talking about his own "calm despair," or the body of his dead friend. The appearance of a dead body is sometimes interpreted in a positive way as "peaceful" and "calm."
The whole poem talks about a calm landscape and a calm ocean. The image is that the ocean is calm as far as one can see, but farther out, in "the deep," it will be heaving. That much makes literal sense. When the wind does not stir up sharp, choppy waves, there are still long, slow, not-very-high waves far from land that are called "heave" or "swell."
"The bounding main" also means the sea. "The main" means the sea and "bounding" means "with waves." I can't imagine how how farms and plains can "mingle" with it. Wait, yes, I can--it's the same image used in Arthur Hugh Clough's poem, "Say Not the Struggle." It probably refers to British coastal landscape somewhere, where the coastline is very tangled and the land is full of dozens of little tidal streams and estuaries connected to the sea.
I'm not familiar enough with old-fashioned nautical terminology to know if "the deep" a specific meaning to sailors. I don't know if sailors use that word nowadays. It's probably a shortening of the phrase "the deep blue sea."
Other similar words that can mean "the sea" are "the depths," "the abyss," and sometimes "blue-water."
Other examples of use:
"Many brave hearts are asleep in the deep, so beware! Beware!" --1897 song about people who have drowned.
"'The Kraken Wakes' is an science fiction novel by John Wyndham, published in the United States in the same year under the title 'Out of the Deeps.'"