You can certainly add minor ticks and show them on the grid but I'm not sure if there's an easy way to label them. In other words, you'll get the tick and the grid lines but not the 1 through 24 labels.
See example below:
% RANDOM DATA - 2 days x 24hrs
StartDate = datetime(2018,1,1,0,0,0);
EndDate = datetime(2018,1,2,23,0,0);
NoOfHours = hours(EndDate-StartDate)+1;
xTimeSpan = StartDate:hours(1):EndDate;
y = rand(1,NoOfHours).*randi([10,50],1,NoOfHours);
f(1) = figure;
ax(1) = gca();
p(1) = plot(xTimeSpan,y,'.k');
grid on
% Grid settings
set(ax(1),...
'XMinorTick','on',...
'XMinorGrid','on',...
'MinorGridColor',[0 1 0],...
'MinorGridAlpha',0.15,...
'MinorGridLineStyle','-');
% x-grid values
set(ax(1).XAxis,...
'TickValues' ,StartDate:EndDate,...
'MinorTickValues',xTimeSpan);
The output:
