Cook onion in butter in a medium nonstick skillet over moderately low heat, stirring, until softened.
Pulse mushrooms in a food processor until coarsely chopped and add to onion along with epazote and salt.
Cook over moderate heat, stirring, until mushrooms are softened and any liquid is evaporated, about 7 minutes. Remove from heat and cool filling completely.
Stir together masa harina, salt, and warm water in a large bowl with a wooden spoon until it forms a dough. Knead dough in bowl with heel of your hand 1 to 2 minutes.
Pinch off small pieces of dough and roll into about 40 (scant 1-inch) balls. Transfer balls as formed to a plastic-wrap–lined tray and cover with more plastic wrap.
Cut 2 (5-inch) squares from a small plastic bag and use 1 to line lower half of tortilla press. Put a ball of dough in tortilla press and top with second square of plastic. Close tortilla press, pushing gently on lever to flatten dough to a 2 1/2- to 3-inch round. If press doesn’t flatten evenly, rotate tortilla 180 degrees and gently flatten again. (Alternatively, use rolling pin to flatten dough between squares of plastic.)
Remove top square of plastic and put 1 teaspoon mushroom filling, or about 1 teaspoon crumbled queso fresco with 1/2 teaspoon crema and a piece of squash blossom, in center of round.
Fold round, still in plastic, over filling to form a half-moon and pinch edges together to seal. Gently peel off plastic, then transfer quesadilla to another plastic-wrap–lined tray and cover with a second sheet of plastic wrap. Repeat with remaining dough, making about 20 quesadillas of each filling.
Fry quesadillas: Heat 1 inch of oil in a deep 10-inch heavy skillet over moderately high heat until deep-fat thermometer registers 375°F, then deep-fry quesadillas in batches of 6 to 8, turning once or twice, until golden, about 4 minutes. Transfer with a slotted spoon to paper towels to drain and return oil to 375°F before adding next batch.
Serve quesadillas immediately.
• Mushroom filling may be made 1 day ahead and chilled, covered.
• Quesadillas may be formed, but not fried, 1 day ahead and chilled, covered. Bring to room temperature, and pat dry if necessary, before frying.