How can I expose functions to the ref using functional components?

While @Matthieu Libeer answer is correct and this should work to you still consider making that in React way. By now your Calendar is the same uncontrolled component. By accessing it's internal state(and methods in case of class-based components) by ref you are loosing:

1. Ability to make pre-input validation(say, to discard selection for each second Sunday in month)
2. Replace it with different component in easy way

Once it would be fully controllable component + keeping consistency with native controls in props' names you would get something like

<Calendar value={currentData} onChange={this.validateAndSetDate} />

This way it can be easily replaced with compatible component(say <input type="date" />) as well as validation is much more easier. You still might need forwardRef say to .focus() imperatively but there will be less pain with other aspects.

Here's how you can do it, even though it's not really the React way. The trick is to set the current value of the ref in Calendar.

const Calendar = React.forwardRef((props, ref) => {
  const [month, setMonth] = useState(0)
  const nextMonth = () => { setMonth((prev) => prev + 1) }
  const prevMonth = () => { setMonth((prev) => prev - 1) }

  ref.current = { nextMonth, prevMonth } // eslint-disable-line

  return (
    <div>{month}</div>
  )
})

const Parent = () => {
  const cal = useRef()

  const onNext = () => {
    cal.current.nextMonth()
  }

  return (
    <React.Fragment>
      <Calendar ref={cal} />
      <button type="button" onClick={onNext}>Next</button>
    </React.Fragment>
  )
}

You can use useImperativeHandle hook.

const Calendar = React.forwardRef((props, ref) => {
  const [month, setMonth] = useState(0)
  const nextMonth = () => { setMonth((prev) => prev + 1) }
  const prevMonth = () => { setMonth((prev) => prev - 1) }

  useImperativeHandle(ref, () => ({
    nextMonth,
    prevMonth
  }));

  return (
    <div>{month}</div>
  )
})

const Parent = () => {
  const cal = useRef()

  const onNext = () => {
    cal.current.nextMonth()
  }

  return (
    <React.Fragment>
      <Calendar ref={cal} />
      <button type="button" onClick={onNext}>Next</button>
    </React.Fragment>
  )
}

in functional components you can use forwardRef to pass ref to child component.

const Calendar = React.forwardRef((props, ref) => {
  const nextMonth = () => {...};
  const prevMonth = () => {...};

  return <div>.....</div>;
});

function Parent() {
  const cal = useRef(null);

  const onNext = () => {
    cal.current.nextMonth();
  };

  return (
    <>
      <Calendar ref={cal} />
      <button onClick={onNext}>Next</button>
    </>
  );
}

this work:

const Calendar = React.forwardRef((props, ref) => {
  const nextMonth = () => console.log("nextMonth");
  const prevMonth = () => console.log("prevMonth");
  ref.current = { nextMonth, prevMonth };
  return <div>.....</div>;
});

function Parent() {
  const cal = useRef(null);

  const onNext = () => {
    cal.current.nextMonth();
  };

  return (
    <>
      <Calendar ref={cal} />
      <button onClick={onNext}>Next</button>
    </>
  );
}