No. The Nintendo GameCube cannot play Nintendo 64 cartridges.
Hardware Incompatibility
Media: N64 uses cartridges; GameCube uses 8 cm miniDVDs. No cartridge slot exists.
Architecture: N64 (MIPS R4300i, 64-bit) vs. GameCube (PowerPC "Gekko", 32-bit). Completely different CPU, GPU, and memory map.
BIOS/OS: No backward-compatibility layer in GameCube firmware.
How to Play N64 Games on GameCube Hardware
Official Ports/Remasters:Zelda: Ocarina of Time / Master Quest (bonus disc), Zelda: Majora's Mask (Collector's Edition), Mario Golf 64 → Mario Golf: Toadstool Tour, etc.
Homebrew (Wii Mode via Wii): A Wii (which plays GameCube games) can run N64 emulators (Not64, Wii64) via Homebrew Channel. GameCube itself has no practical N64 emulation.
Nintendo Switch Online + Expansion Pack: Legal N64 library on modern hardware.