What's the best way to draw a fullscreen quad in OpenGL 3.2?

You can send two triangles creating a quad, with their vertex attributes set to -1/1 respectively.

You do not need to multiply them with any matrix in the vertex/fragment shader.

Here are some code samples, simple as it is :)

Vertex Shader:

const vec2 madd=vec2(0.5,0.5);
attribute vec2 vertexIn;
varying vec2 textureCoord;
void main() {
   textureCoord = vertexIn.xy*madd+madd; // scale vertex attribute to [0-1] range
   gl_Position = vec4(vertexIn.xy,0.0,1.0);
}

Fragment Shader :

varying vec2 textureCoord;
void main() {
   vec4 color1 = texture2D(t,textureCoord);
   gl_FragColor = color1;
}

To output a fullscreen quad geometry shader can be used:

#version 330 core

layout(points) in;
layout(triangle_strip, max_vertices = 4) out;

out vec2 texcoord;

void main() 
{
    gl_Position = vec4( 1.0, 1.0, 0.5, 1.0 );
    texcoord = vec2( 1.0, 1.0 );
    EmitVertex();

    gl_Position = vec4(-1.0, 1.0, 0.5, 1.0 );
    texcoord = vec2( 0.0, 1.0 ); 
    EmitVertex();

    gl_Position = vec4( 1.0,-1.0, 0.5, 1.0 );
    texcoord = vec2( 1.0, 0.0 ); 
    EmitVertex();

    gl_Position = vec4(-1.0,-1.0, 0.5, 1.0 );
    texcoord = vec2( 0.0, 0.0 ); 
    EmitVertex();

    EndPrimitive(); 
}

Vertex shader is just empty:

#version 330 core

void main()
{
}

To use this shader you can use dummy draw command with empty VBO:

glDrawArrays(GL_POINTS, 0, 1);

No need to use a geometry shader, a VBO or any memory at all.

A vertex shader can generate the quad.

layout(location = 0) out vec2 uv;

void main() 
{
    float x = float(((uint(gl_VertexID) + 2u) / 3u)%2u); 
    float y = float(((uint(gl_VertexID) + 1u) / 3u)%2u); 

    gl_Position = vec4(-1.0f + x*2.0f, -1.0f+y*2.0f, 0.0f, 1.0f);
    uv = vec2(x, y);
}

Bind an empty VAO. Send a draw call for 6 vertices.