BUG: DataFrame.convert_dtypes() drops the columns' name
- [x] I have checked that this issue has not already been reported.

- [x] I have confirmed this bug exists on the latest version of pandas.

- [ ] (optional) I have confirmed this bug exists on the master branch of pandas.

#### Code Sample, a copy-pastable example

```python
import pandas as pd


df = pd.DataFrame({'a': [1, 2], 'b': [3, 4]})
df.columns.name = 'cols'
print(df)
# drops the columns' name 'cols'
print(df.convert_dtypes())

```

#### Problem description

DataFrame.convert_dtypes() drops the columns' name.
This is unexpected.

#### Expected Output

convert_dtypes() should keep the columns' name.
It does so for the index name.

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit           : 2cb96529396d93b46abab7bbc73a208e708c642e
python           : 3.8.8.final.0
python-bits      : 64
OS               : Linux
OS-release       : 5.8.0-50-generic
Version          : #56~20.04.1-Ubuntu SMP Mon Apr 12 21:46:35 UTC 2021
machine          : x86_64
processor        : x86_64
byteorder        : little
LC_ALL           : None
LANG             : en_US.UTF-8
LOCALE           : en_US.UTF-8

pandas           : 1.2.4
numpy            : 1.20.1
pytz             : 2021.1
dateutil         : 2.8.1
pip              : 21.0.1
setuptools       : 52.0.0.post20210125
Cython           : None
pytest           : None
hypothesis       : None
sphinx           : None
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : None
pymysql          : None
psycopg2         : None
jinja2           : None
IPython          : None
pandas_datareader: None
bs4              : None
bottleneck       : None
fsspec           : 0.9.0
fastparquet      : None
gcsfs            : None
matplotlib       : 3.3.4
numexpr          : None
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : 1.0.0
pyxlsb           : None
s3fs             : 0.6.0
scipy            : 1.6.2
sqlalchemy       : None
tables           : None
tabulate         : None
xarray           : None
xlrd             : None
xlwt             : None
numba            : None

</details>
