Rosh Chodesh Tishrey

So this period like today we don't bless and don't say Chodesh Tov?

ב"ה Shalom Well, generally people bless each other by saying Chodesh Tov in the beginning of the month. The month of Elul is like any other month from that point of view, and people do say Chodesh Tov. It is only the month of Tishrei, which we don't bless Rosh Chodesh in advance like the other months for the reasons I gave in a previous answer. Then, when Tishrei approaches, people usually bless each other for a good year and not just a month. Not that it's forbidden, but that is what is simply done. Also, after Kiddush Levana of Tishrie, which by most Ashkenazim is done at the end of Yom Kippur, people bless each other with Chodesh Tov. All the Best and Shana Tova