How can I change the text box on a surface gui on a brick, with a gui?

There is a event of TextBox objects called FocusLost. This event fires after the player who clicked on the GUI presses enter or clicks away from the GUI. In you're case, I'm going to assume that you want to make sure that the player presses enter in order to transfer over the text.

(Make sure this is in a local script)

local screenTextBox = script.Parent -- Location of Textbox on the screen
local partTextLabel = game.Workspace.Sign.SurfaceGui.TextLabel -- Location of text on block

screenTextBox.FocusLost:connect(function(enterPressed)
    if enterPressed then -- Make sure they loose focus due to enter press
        partTextLabel.Text = screenTextBox.Text -- Change text
    end
end)