The offer was as enticing as it was unexpected for a relative political newcomer. Three years ago, Prime Minister Justin Trudeau asked Mélanie Joly to become foreign affairs minister, among the most prestigious and highest profile portfolios in Canada’s cabinet.

But Ms. Joly, who at the time held a significantly less influential ministerial role, turned him down flat.

Her refusal wasn’t because of the fact that she lacked foreign policy experience. She said no because she feared that the travel involved in the globe-trotting job would force her to abandon her yearslong quest to conceive a child through in vitro fertilization.

But Canada’s leader refused to give up.

Mr. Trudeau offered to make whatever arrangements necessary to maintain Ms. Joly’s treatment anywhere in the world. “‘If you become pregnant,’” she remembers him telling her, “‘it would be a fantastic message you would send to the world.’”

After consulting her husband, parents and siblings, Ms. Joly relented, becoming Canada’s top diplomat.