Third Monitor is not detecting

For a 14" MacBook Pro with a M3 Pro chip, the Technical Specifications say that it

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Simultaneously supports full native resolution on the built-in display at 1 billion colors and:

• Up to two external displays with up to 6K resolution at 60Hz over Thunderbolt, or one external display with up to 6K resolution at 60Hz over Thunderbolt and one external display with up to 4K resolution at 144Hz over HDMI
• One external display supported at 8K resolution at 60Hz or one external display at 4K resolution at 240Hz over HDMI

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So just to get things straight, when you said "third monitor", were you referring to a second external monitor (two external monitors + the built-one = three)?

If you are trying to connect two monitors to a M3-Pro-based machine, through a Thunderbolt dock (not merely a USB-C dock marketed as "Thunderbolt-compatible"), that should be possible. At least with some docks – since Other World Computing, SonnetTech, and CalDigit offer many Thunderbolt docks which they say can do that, in one way or another.

If you are trying to connect three monitors to a M3-Pro-based machine, through a Thunderbolt dock, that's more monitors than the M3 Pro chip can support.

What type of chip is in that 14" MacBook Pro? Is it a base M3 chip, a M3 Pro, or a M3 Max?

And is that a Thunderbolt docking station, a plain USB-C one, or something else?