There are so many situations in which we might hear the noun quadro. Let's look at some of the most common ones.
The first meaning of quadro has to do with shape. Un quadro (a square) has quattro lati uguali (four equal sides) so we can see the relation between quadro and quattro.
We use the adjective quadrato to mean "square." Sometimes quadro and quadrato can be interchangeable both as nouns and as adjectives. When we talk about measurements, it's common to see either metri quadri or metri quadrati, which both mean "square meters." A common abbreviation is mq. With kilometers it's more common to see chilometri quadrati (square kilometers).
Si sviluppava il castello su una superficie di undici mila metri quadri.
The castle was built over an area of eleven thousand square meters.
Caption 33, Escursioni Campane - Castello Normanno - Part 1Play Caption
L'isola di Vulcano, con i suoi ventuno chilometri quadrati di superficie,
è la terza fra le sette sorelle delle isole Eolie.
The island of Vulcano, with its twenty-one square kilometers of surface area, is the third among the seven sisters of the Aeolian Islands.
Captions 1-2, Linea Blu - Le Eolie - Part 16Play Caption
One reason we might use quadrato as a noun to mean "a square," rather than quadro, is because it's unambiguous. Un quadrato is a square, no doubt about it.
Disegniamo un quadrato nel centro del foglio (Let's draw a square in the middle of the page).
Un quadro, on the other hand, can mean "a painting," so when talking about art, it's wise to distinguish. Paintings are usually on a canvas, and the canvas is usually four-sided (admittedly, not always square).
I quadri — paintings can be of different types: un ritratto (a portrait) or a scene. And sometimes quadro stands for "scene," as in the theater for example.
Turandot, atto terzo, quadro primo.
Turandot, third act, scene one.
Caption 15, La Ladra - Ep. 2 - Viva le spose - Part 1Play Caption
Another very different meaning for quadro is "control panel." This can be in a vehicle, as in the following example, or quadro can describe the fuse box, or eletrical switchboard.
Ci sono ancora le chiavi attaccate al quadro. -Sì.
The keys are still in the ignition. -Yes.
E qualcuno è andato in giro con questa macchina fino all'una.
And someone went around with this car until one o'clock.
Captions 32-33, Il Commissario Manara - S1EP10 - Un morto di troppo - Part 5Play Caption
There are other meanings and sfumature (nuances) for the word quadro, and Marika talks about one of them here.