Is it true that late in the war all German infantry divisions were designated Panzer-Grenadier Divisions in order to boost morale?

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Cheers,
Daniel

No, not all infantry divisions were re-designated Panzergrenadier divisions. Motorisierte —i.e. motorised— divisions were re-designated Panzergrenadier.

And many infantry divisions were up-graded and at least in theory, motorised.

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"War is less costly than servitude, the choice is always between Verdun and Dachau." - Jean Dutourd, French veteran of both world wars

"A mon fils: depuis que tes yeux sont fermes les miens n’ont cessé de pleurir." - Mère française, Verdun

Also, for morale effects some Infantry divisions were re-designated "Volksgrenadier"

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Retreat!...Hell,we just get here!